3.3.35 \(\int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx\) [235]

Optimal. Leaf size=126 \[ \frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {2 a c \cos ^3(e+f x)}{315 f \left (c^2-c^2 \sin (e+f x)\right )^3} \]

[Out]

1/9*a*c*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^6+1/21*a*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^5+2/105*a*cos(f*x+e)^3/c/f/(c
-c*sin(f*x+e))^4+2/315*a*c*cos(f*x+e)^3/f/(c^2-c^2*sin(f*x+e))^3

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Rubi [A]
time = 0.15, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2815, 2751, 2750} \begin {gather*} \frac {2 a c \cos ^3(e+f x)}{315 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]

[Out]

(a*c*Cos[e + f*x]^3)/(9*f*(c - c*Sin[e + f*x])^6) + (a*Cos[e + f*x]^3)/(21*f*(c - c*Sin[e + f*x])^5) + (2*a*Co
s[e + f*x]^3)/(105*c*f*(c - c*Sin[e + f*x])^4) + (2*a*c*Cos[e + f*x]^3)/(315*f*(c^2 - c^2*Sin[e + f*x])^3)

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 2815

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))

Rubi steps

\begin {align*} \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx &=(a c) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {1}{3} a \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {(2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^4} \, dx}{21 c}\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {(2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{105 c^2}\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {2 a \cos ^3(e+f x)}{315 c^2 f (c-c \sin (e+f x))^3}\\ \end {align*}

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Mathematica [A]
time = 0.40, size = 124, normalized size = 0.98 \begin {gather*} \frac {a \left (315 \cos \left (e+\frac {f x}{2}\right )-84 \cos \left (e+\frac {3 f x}{2}\right )+9 \cos \left (3 e+\frac {7 f x}{2}\right )+189 \sin \left (\frac {f x}{2}\right )+36 \sin \left (2 e+\frac {5 f x}{2}\right )-\sin \left (4 e+\frac {9 f x}{2}\right )\right )}{1260 c^5 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]

[Out]

(a*(315*Cos[e + (f*x)/2] - 84*Cos[e + (3*f*x)/2] + 9*Cos[3*e + (7*f*x)/2] + 189*Sin[(f*x)/2] + 36*Sin[2*e + (5
*f*x)/2] - Sin[4*e + (9*f*x)/2]))/(1260*c^5*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9)

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Maple [A]
time = 0.39, size = 146, normalized size = 1.16

method	result	size
risch	\(-\frac {4 i a \left (189 i {\mathrm e}^{4 i \left (f x +e \right )}+315 \,{\mathrm e}^{5 i \left (f x +e \right )}+36 i {\mathrm e}^{2 i \left (f x +e \right )}-84 \,{\mathrm e}^{3 i \left (f x +e \right )}-i+9 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{315 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9}}\)	\(85\)
derivativedivides	\(\frac {2 a \left (-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {236}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {248}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {148}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {32}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\)	\(146\)
default	\(\frac {2 a \left (-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {236}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {248}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {148}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {32}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\)	\(146\)
norman	\(\frac {\frac {6 a \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {116 a}{315 c f}-\frac {2 a \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {28 a \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {616 a \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {56 a \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {46 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 c f}+\frac {544 a \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {1108 a \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}+\frac {1636 a \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}-\frac {2402 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(241\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

2/f*a/c^5*(-16/(tan(1/2*f*x+1/2*e)-1)^8-236/5/(tan(1/2*f*x+1/2*e)-1)^5-5/(tan(1/2*f*x+1/2*e)-1)^2-248/7/(tan(1
/2*f*x+1/2*e)-1)^7-1/(tan(1/2*f*x+1/2*e)-1)-148/3/(tan(1/2*f*x+1/2*e)-1)^6-32/9/(tan(1/2*f*x+1/2*e)-1)^9-32/(t
an(1/2*f*x+1/2*e)-1)^4-46/3/(tan(1/2*f*x+1/2*e)-1)^3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (130) = 260\).
time = 0.32, size = 799, normalized size = 6.34 \begin {gather*} -\frac {2 \, {\left (\frac {a {\left (\frac {432 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {1728 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3612 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {5418 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {5040 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {3360 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {1260 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {315 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 83\right )}}{c^{5} - \frac {9 \, c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {84 \, c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {126 \, c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, c^{5} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {36 \, c^{5} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}} - \frac {5 \, a {\left (\frac {45 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {117 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {273 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {315 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {147 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - 5\right )}}{c^{5} - \frac {9 \, c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {84 \, c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {126 \, c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, c^{5} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {36 \, c^{5} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}}\right )}}{315 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

-2/315*(a*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3612*sin(f*x + e)^
3/(cos(f*x + e) + 1)^3 - 5418*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5040*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 -
3360*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1260*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 315*sin(f*x + e)^8/(cos
(f*x + e) + 1)^8 - 83)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)
^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*
x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x +
e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) - 5*a*(45*sin
(f*x + e)/(cos(f*x + e) + 1) - 117*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 273*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 - 315*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 147*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 + 63*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1)
+ 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x
+ e)^9/(cos(f*x + e) + 1)^9))/f

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (130) = 260\).
time = 0.33, size = 276, normalized size = 2.19 \begin {gather*} -\frac {2 \, a \cos \left (f x + e\right )^{5} - 8 \, a \cos \left (f x + e\right )^{4} - 25 \, a \cos \left (f x + e\right )^{3} + 20 \, a \cos \left (f x + e\right )^{2} - 35 \, a \cos \left (f x + e\right ) + {\left (2 \, a \cos \left (f x + e\right )^{4} + 10 \, a \cos \left (f x + e\right )^{3} - 15 \, a \cos \left (f x + e\right )^{2} - 35 \, a \cos \left (f x + e\right ) - 70 \, a\right )} \sin \left (f x + e\right ) - 70 \, a}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

-1/315*(2*a*cos(f*x + e)^5 - 8*a*cos(f*x + e)^4 - 25*a*cos(f*x + e)^3 + 20*a*cos(f*x + e)^2 - 35*a*cos(f*x + e
) + (2*a*cos(f*x + e)^4 + 10*a*cos(f*x + e)^3 - 15*a*cos(f*x + e)^2 - 35*a*cos(f*x + e) - 70*a)*sin(f*x + e) -
70*a)/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c
^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^
5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1700 vs. \(2 (112) = 224\).
time = 13.11, size = 1700, normalized size = 13.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((-630*a*tan(e/2 + f*x/2)**8/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 1134
0*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**
5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*ta
n(e/2 + f*x/2) - 315*c**5*f) + 1890*a*tan(e/2 + f*x/2)**7/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/
2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f
*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)
**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 5250*a*tan(e/2 + f*x/2)**6/(315*c**5*f*tan(e/2 + f*x/2)**9
- 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 3969
0*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**
5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 6930*a*tan(e/2 + f*x/2)**5/(315*c**5*f*
tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/
2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f
*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 7686*a*tan(e/2 + f*
x/2)**4/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 -
26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 2646
0*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) +
4494*a*tan(e/2 + f*x/2)**3/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*t
an(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/
2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*
x/2) - 315*c**5*f) - 2286*a*tan(e/2 + f*x/2)**2/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)
**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 -
39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835
*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 414*a*tan(e/2 + f*x/2)/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*
tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e
/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 +
f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 116*a/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*t
an(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/
2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f
*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f), Ne(f, 0)), (x*(a*sin(e) + a)/(-c*sin(e) + c)**5, True))

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Giac [A]
time = 0.43, size = 144, normalized size = 1.14 \begin {gather*} -\frac {2 \, {\left (315 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 945 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2625 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3465 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3843 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2247 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1143 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 207 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 58 \, a\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-2/315*(315*a*tan(1/2*f*x + 1/2*e)^8 - 945*a*tan(1/2*f*x + 1/2*e)^7 + 2625*a*tan(1/2*f*x + 1/2*e)^6 - 3465*a*t
an(1/2*f*x + 1/2*e)^5 + 3843*a*tan(1/2*f*x + 1/2*e)^4 - 2247*a*tan(1/2*f*x + 1/2*e)^3 + 1143*a*tan(1/2*f*x + 1
/2*e)^2 - 207*a*tan(1/2*f*x + 1/2*e) + 58*a)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)

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Mupad [B]
time = 8.77, size = 119, normalized size = 0.94 \begin {gather*} \frac {\sqrt {2}\,a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {121\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {1575\,\sin \left (e+f\,x\right )}{4}-\frac {625\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {635\,\cos \left (e+f\,x\right )}{4}+\frac {7\,\cos \left (4\,e+4\,f\,x\right )}{2}+\frac {399\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {141\,\sin \left (3\,e+3\,f\,x\right )}{4}-\frac {15\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {1357}{4}\right )}{5040\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))/(c - c*sin(e + f*x))^5,x)

[Out]

(2^(1/2)*a*cos(e/2 + (f*x)/2)*((121*cos(3*e + 3*f*x))/4 - (1575*sin(e + f*x))/4 - (625*cos(2*e + 2*f*x))/4 - (
635*cos(e + f*x))/4 + (7*cos(4*e + 4*f*x))/2 + (399*sin(2*e + 2*f*x))/4 + (141*sin(3*e + 3*f*x))/4 - (15*sin(4
*e + 4*f*x))/4 + 1357/4))/(5040*c^5*f*cos(e/2 + pi/4 + (f*x)/2)^9)

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